The autocorrelation function of a Gaussian random process representing the unevenness of a road surface is given by

\[R_{x}(\tau)=\sigma_{x}^{2} e^{-\alpha|v \tau|} \cos \beta v \tau\]

where \(\sigma_{x}^{2}\) is the variance of the random process and \(v\) is the velocity of the vehicle. The values of \(\sigma_{x}, \alpha\), and \(\beta\) for different types of road are as follows:

Compute the spectral density of the road surface for the different types of road.

Type of Road a Asphalt 1.1 0.2 0.4 Paved 1.6 0.3 0.6 Gravel 1.8 0.5 0.9